The proof of [1]} produces a (not necessarily ergodic) \(G\) -system satisfying (REF ). We will repeat the proof therein with one modification: begin by choosing an extreme invariant mean on \(\ell ^\infty (G)\)  with \(m(1_A)=d^*(A)\) . Define the system \((X,\mu ,T)\)  as follows

\(X:=\lbrace 0,1\rbrace ^G\)  with the product topology, meaning \(X\)  is the space of all functions from \(G\)  into \(\lbrace 0,1\rbrace \) , with the topology of pointwise convergence,

\(T_g\)  is defined by \(T_g x(h)= x(h+g)\) ,

and \(\mu \)  is defined to satisfy
\(\int f\, d\mu = m(\phi _f)\) 
for every continuous \(f:X\rightarrow \mathbb {C}\) , where \(\phi _f:G\rightarrow \mathbb {C}\)  is given by \(\phi _f(g)=f(1_{A+g})\) .

Having defined \(\int f\, d\mu \)  for continuous \(f\) , the Riesz representation theorem shows that there really is a Borel measure \(\mu \)  satisfying this definition. We now show that our definition of \(\mu \)  agrees with the definition of \(\mu \)  in [1]}. We do so by showing that the two definitions agree on the cylinder sets in \(X\) . Writing such a cylinder set \(C\)  by fixing \(h_1,\dots ,h_k\in G\)  and setting \(C:=\lbrace x\in X: x(h_1)=\varepsilon _1,\dots , x(h_k)=\varepsilon _k\rbrace \) , we get that \(\phi _f(g)=1\)  if and only if \(g\in (A_1-h_1)\cap (A_2-h_2)\cap \dots \cap (A_k-h_k)\) , where \(A_i=A\)  if \(\varepsilon _i=1\)  and \(G\setminus A\)  if \(\varepsilon _i=0\) . Thus \(\mu (C) = m(\phi _f)=m(\prod _{i=1}^k 1_{A_i-h_i})\) , and the latter is the definition of \(\mu (C)\)  in [1]}. Following the argument in [1]} proves inequality (REF ).
To prove ergodicity of \((X,\mu ,T)\)  we will prove that for all \(f_1, f_2\in L^\infty (\mu )\) , we have
\(\lim _{n\rightarrow \infty } \frac{1}{|F_n|}\sum _{g\in F_n} \int f_1\cdot T_gf_2\, d\mu =\int f_1\, d\mu \int f_2\, d\mu \) 
for each Følner sequence \((F_n)_{n\in \mathbb {N}}\) . To see this, note first that it suffices to prove this identity for continuous \(f_i\) , by approximating bounded \(f_i\)  with continuous functions. Now choose an invariant mean \(\eta \)  on \(\ell ^\infty (G)\)  with the property that \(\lim _{n\rightarrow \infty } \frac{1}{|F_n|}\sum _{g\in F_n}\phi (g)=\eta (\phi )\)  whenever the limit on the left exists. With this choice of \(\eta \) , the definition of \(\mu \)  lets us write the left-hand side of (REF ) as
\(\int \phi _{f_1}(h) \phi _{f_2}(h+g)\, dm(h)\, d\eta (g)\) 
and the extremality of \(m\)  allows us to simplify this as \(m(\phi _{f_1})m(\phi _{f_2})\) , by (REF ). By the definition of \(\mu \) , this is just \(\int f_1\,d\mu \int f_2\, d\mu \) , as desired.
Now (REF ) implies ergodicity of \((X,\mu ,T)\) : if \(A\)  is \(T\) -invariant, then setting \(f_1=f_2=1_A\)  in (REF ), we get \(\mu (A)=\mu (A)^2\) , which means \(\mu (A)=0\)  or 1.
